The data comes from R’s built-in dataset called mtcars. Each observation consists of 11 features describing various parameters of selected car models. The aim of this project is to perform both supervised and unsupervised classification and to describe each variable.

Analysis Plan:

Presentation of summary statistics including histograms
Univariate analysis of outliers and multimodality
Correlation analysis and dimensionality reduction
Clustering analysis
Supervised classification using random forests

cat('Basic statistical measures','\n')

## Basic statistical measures

cat('\n')

apply(mtcars[,], 2 ,summary)

##              mpg    cyl     disp       hp     drat      wt     qsec     vs
## Min.    10.40000 4.0000  71.1000  52.0000 2.760000 1.51300 14.50000 0.0000
## 1st Qu. 15.42500 4.0000 120.8250  96.5000 3.080000 2.58125 16.89250 0.0000
## Median  19.20000 6.0000 196.3000 123.0000 3.695000 3.32500 17.71000 0.0000
## Mean    20.09062 6.1875 230.7219 146.6875 3.596563 3.21725 17.84875 0.4375
## 3rd Qu. 22.80000 8.0000 326.0000 180.0000 3.920000 3.61000 18.90000 1.0000
## Max.    33.90000 8.0000 472.0000 335.0000 4.930000 5.42400 22.90000 1.0000
##              am   gear   carb
## Min.    0.00000 3.0000 1.0000
## 1st Qu. 0.00000 3.0000 2.0000
## Median  0.00000 4.0000 2.0000
## Mean    0.40625 3.6875 2.8125
## 3rd Qu. 1.00000 4.0000 4.0000
## Max.    1.00000 5.0000 8.0000

cat('\n')

cat('Standard deviation','\n')

## Standard deviation

cat('\n')

apply(mtcars[,] , 2 , sd)

##         mpg         cyl        disp          hp        drat          wt 
##   6.0269481   1.7859216 123.9386938  68.5628685   0.5346787   0.9784574 
##        qsec          vs          am        gear        carb 
##   1.7869432   0.5040161   0.4989909   0.7378041   1.6152000

cat('\n')

cat('Skewness','\n')

## Skewness

cat('\n')

apply(mtcars[,] , 2 ,skewness)

##        mpg        cyl       disp         hp       drat         wt       qsec 
##  0.6404399 -0.1831287  0.4002724  0.7614356  0.2788734  0.4437855  0.3870456 
##         vs         am       gear       carb 
##  0.2519763  0.3817709  0.5546495  1.1021304

cat('\n')

cat('Kurtosis','\n')

## Kurtosis

cat('\n')

apply(mtcars[,] , 2 ,kurtosis)

##      mpg      cyl     disp       hp     drat       wt     qsec       vs 
## 2.799467 1.319032 1.910317 3.052233 2.435116 3.172471 3.553753 1.063492 
##       am     gear     carb 
## 1.145749 2.056790 4.536121

Summary and Conclusions:

The variables disp and hp have the highest minimum values and also the highest mean and median. They also have high standard deviation values, indicating low concentration around the mean. The rest of the variables show low standard deviation, suggesting high concentration around the mean. All variables have kurtosis greater than 0, indicating leptokurtic distributions. The variable carb has the highest kurtosis and the strongest presence of extreme values.

for(i in 1:11) print(grubbs.test(mtcars[,i]))

## 
##  Grubbs test for one outlier
## 
## data:  mtcars[, i]
## G = 2.29127, U = 0.82518, p-value = 0.276
## alternative hypothesis: highest value 33.9 is an outlier
## 
## 
##  Grubbs test for one outlier
## 
## data:  mtcars[, i]
## G = 1.22486, U = 0.95004, p-value = 1
## alternative hypothesis: lowest value 4 is an outlier
## 
## 
##  Grubbs test for one outlier
## 
## data:  mtcars[, i]
## G = 1.9468, U = 0.8738, p-value = 0.7363
## alternative hypothesis: highest value 472 is an outlier
## 
## 
##  Grubbs test for one outlier
## 
## data:  mtcars[, i]
## G = 2.74657, U = 0.74881, p-value = 0.05564
## alternative hypothesis: highest value 335 is an outlier
## 
## 
##  Grubbs test for one outlier
## 
## data:  mtcars[, i]
## G = 2.4939, U = 0.7929, p-value = 0.1419
## alternative hypothesis: highest value 4.93 is an outlier
## 
## 
##  Grubbs test for one outlier
## 
## data:  mtcars[, i]
## G = 2.25534, U = 0.83063, p-value = 0.3083
## alternative hypothesis: highest value 5.424 is an outlier
## 
## 
##  Grubbs test for one outlier
## 
## data:  mtcars[, i]
## G = 2.82675, U = 0.73393, p-value = 0.04021
## alternative hypothesis: highest value 22.9 is an outlier
## 
## 
##  Grubbs test for one outlier
## 
## data:  mtcars[, i]
## G = 1.11604, U = 0.95853, p-value = 1
## alternative hypothesis: highest value 1 is an outlier
## 
## 
##  Grubbs test for one outlier
## 
## data:  mtcars[, i]
## G = 1.18990, U = 0.95285, p-value = 1
## alternative hypothesis: highest value 1 is an outlier
## 
## 
##  Grubbs test for one outlier
## 
## data:  mtcars[, i]
## G = 1.77893, U = 0.89462, p-value = 1
## alternative hypothesis: highest value 5 is an outlier
## 
## 
##  Grubbs test for one outlier
## 
## data:  mtcars[, i]
## G = 3.21168, U = 0.65653, p-value = 0.006787
## alternative hypothesis: highest value 8 is an outlier

Conclusions:

For the variables qsec and carb, the Grubbs test p-value is less than 0.05, indicating the presence of outliers that must be considered during clustering analysis.

for(i in 1:10) hist(mtcars[,i+1], main=colnames(mtcars)[i+1])

Conclusions:

Histogram analysis suggests that normality should be tested for hp and drat, and multimodality for disp and wt.

Normality Tests:

shapiro.test(mtcars$hp)

## 
##  Shapiro-Wilk normality test
## 
## data:  mtcars$hp
## W = 0.93342, p-value = 0.04881

shapiro.test(mtcars$drat)

## 
##  Shapiro-Wilk normality test
## 
## data:  mtcars$drat
## W = 0.94588, p-value = 0.1101

Conclusions:

The hypothesis of normality for variables hp and drat is rejected.

Multimodality Analysis:

silverman.test(mtcars$disp,k=1)

## Silvermantest: Testing the null hypothesis that the number of modes is <=  1 
## The resulting p-value is  0.1781782

silverman.test(mtcars$wt,k=1)

## Silvermantest: Testing the null hypothesis that the number of modes is <=  1 
## The resulting p-value is  0.08608609

Conclusions:

The distributions of disp and wt are not multimodal, as the significance level exceeds 5%.

Correlation Analysis using Pearson’s coefficient

heatmaply_cor(cor(mtcars[,],method='pearson'))

Conclusions:

Positive correlations are observed within the groups: (wt, disp, cyl, hp), (gear, am, drat), and (vs, qsec). A weaker positive correlation is present between vs and carb.

Correlation Analysis using Kendall’s coefficient

heatmaply_cor(cor(mtcars[,],method='kendall'))

Conclusions:

The variables are less correlated in the same areas as with Pearson’s method.

PCA

prcomp(mtcars[,-1])->pca.mtcars
summary(pca.mtcars)

## Importance of components:
##                             PC1      PC2     PC3     PC4     PC5     PC6    PC7
## Standard deviation     136.4339 38.14648 1.31542 0.96553 0.76789 0.32620 0.2892
## Proportion of Variance   0.9273  0.07249 0.00009 0.00005 0.00003 0.00001 0.0000
## Cumulative Proportion    0.9273  0.99982 0.99991 0.99995 0.99998 0.99999 1.0000
##                           PC8   PC9   PC10
## Standard deviation     0.2508 0.222 0.1999
## Proportion of Variance 0.0000 0.000 0.0000
## Cumulative Proportion  1.0000 1.000 1.0000

pca.mtcars$rotation[,1]

##          cyl         disp           hp         drat           wt         qsec 
##  0.012042615  0.900235270  0.435074057 -0.002661394  0.006242550 -0.006676533 
##           vs           am         gear         carb 
## -0.002731293 -0.001963245 -0.002606103  0.005767541

df.pca=data.frame(pc1=pca.mtcars$x[,1],pc2=pca.mtcars$x[,2],pc3=pca.mtcars$x[,3],kl=as.factor(mtcars$cyl))
plot_ly(df.pca,x=~pc1,y=~pc2,z=~pc3,color = ~kl,type='scatter3d')

## No scatter3d mode specifed:
##   Setting the mode to markers
##   Read more about this attribute -> https://plotly.com/r/reference/#scatter-mode

Conclusions:

The first principal component explains 92.73% of the total variance. The variable am has the strongest influence on PC1. Clear cluster separation is visible along the PC1 axis.

Unsupervised classification using k-means clustering

kmeans(mtcars[,-1], centers=3)->km.mtcars.3
table(km.mtcars.3$cluster,mtcars$cyl)

##    
##      4  6  8
##   1 11  5  0
##   2  0  2  5
##   3  0  0  9

df.km<-data.frame(x=pca.mtcars$x[,1],y=pca.mtcars$x[,2],z=pca.mtcars$x[,3],type=as.factor(km.mtcars.3$cluster))
plot_ly(df.km,x=~x,y=~y, z=~z,color=~type,type ='scatter3d')

## No scatter3d mode specifed:
##   Setting the mode to markers
##   Read more about this attribute -> https://plotly.com/r/reference/#scatter-mode

Conclusions:

A clear division into 3 clusters is visible, with one of the clusters being the most dispersed.

Determining the optimal number of clusters

wss<-NA
for(i in 1:11) wss<-c(wss,kmeans(mtcars[,-1], centers=i)$tot.withinss)
plot(1:length(wss),wss)

Conclusion:

Using the elbow method, it can be concluded that 3 or 4 clusters are most optimal.

plot(agnes(dist(mtcars[,],method='minkowski',p=1),diss=T))

plot(diana(mtcars[,]))

Conclusions:

Both methods yield similar results. The coefficients differ by only 0.02.

Supervised Classification

We use the random forest method.

rfcv(mtcars[,-2],as.factor(mtcars$cyl))->rf.ic
rf.ic$error.cv

##      10       5       2       1 
## 0.06250 0.06250 0.09375 0.09375

plot(rf.ic$n.var, rf.ic$error.cv, type="b", 
     xlab="Number of variables", ylab="Classification error", 
     main="Optimal number of variables")

Conclusion:

The smallest cross-validation error occurs when the number of variables is between 2 and 5.

randomForest(mtcars[,-2],as.factor(mtcars$cyl),importance = T)->rf.mtcars.imp
varImpPlot(rf.mtcars.imp)

Conclusions:

According to the accuracy drop criterion, the most important variables are disp, mpg, hp, and wt. This number of variables is acceptable based on previous results.

randomForest(mtcars[,c(3,1,4,6)],as.factor(mtcars$cyl))->rf.mtcars.sel
rf.mtcars.sel$confusion

##    4 6  8 class.error
## 4 10 1  0  0.09090909
## 6  1 5  1  0.28571429
## 8  0 0 14  0.00000000

Conclusion:

Using these variables, the maximum classification error is 28%.

Final Conclusions

The variables disp and hp have high mean and standard deviation values. The Grubbs test indicated outliers in qsec and carb, which should be considered in cluster analysis. Histogram analysis helped identify variables for normality and multimodality testing. Normality tests showed that hp and drat do not follow a normal distribution, while Silverman’s test did not confirm multimodality for disp and wt. PCA analysis showed that the first principal component (PC1) explains 92.73% of the variance, mainly influenced by am. K-means clustering and the elbow method indicated 3 or 4 optimal clusters. Random forests are not an optimal classification model in this case, as the maximum classification error is 28%.

Mtcars analysis

Wojciech Kolasa

2025-07-23

Analysis Plan:

Summary and Conclusions:

Conclusions:

Conclusions:

Normality Tests:

Conclusions:

Multimodality Analysis:

Conclusions:

Correlation Analysis using Pearson’s coefficient

Conclusions:

Correlation Analysis using Kendall’s coefficient

Conclusions:

PCA

Conclusions:

Unsupervised classification using k-means clustering

Conclusions:

Determining the optimal number of clusters

Conclusion:

Conclusions:

Supervised Classification

Conclusion:

Conclusions:

Conclusion:

Final Conclusions